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Theorem nsspssun 4000
Description: Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)
Assertion
Ref Expression
nsspssun 𝐴𝐵𝐵 ⊊ (𝐴𝐵))

Proof of Theorem nsspssun
StepHypRef Expression
1 ssun2 3920 . . . 4 𝐵 ⊆ (𝐴𝐵)
21biantrur 528 . . 3 (¬ (𝐴𝐵) ⊆ 𝐵 ↔ (𝐵 ⊆ (𝐴𝐵) ∧ ¬ (𝐴𝐵) ⊆ 𝐵))
3 ssid 3765 . . . . 5 𝐵𝐵
43biantru 527 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵𝐵𝐵))
5 unss 3930 . . . 4 ((𝐴𝐵𝐵𝐵) ↔ (𝐴𝐵) ⊆ 𝐵)
64, 5bitri 264 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) ⊆ 𝐵)
72, 6xchnxbir 322 . 2 𝐴𝐵 ↔ (𝐵 ⊆ (𝐴𝐵) ∧ ¬ (𝐴𝐵) ⊆ 𝐵))
8 dfpss3 3835 . 2 (𝐵 ⊊ (𝐴𝐵) ↔ (𝐵 ⊆ (𝐴𝐵) ∧ ¬ (𝐴𝐵) ⊆ 𝐵))
97, 8bitr4i 267 1 𝐴𝐵𝐵 ⊊ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 383  cun 3713  wss 3715  wpss 3716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-v 3342  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731
This theorem is referenced by:  disjpss  4172  lindsenlbs  33717
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